# sage.doctest: needs sage.libs.singular
"""
Base class of curves

This module defines the base class of curves in Sage.

Curves in Sage are reduced subschemes of dimension 1 of an ambient space. The ambient space
is either an affine space or a projective space.

EXAMPLES::

    sage: A.<x,y,z> = AffineSpace(QQ, 3)
    sage: C = Curve([x - y, z - 2])
    sage: C
    Affine Curve over Rational Field defined by x - y, z - 2
    sage: C.dimension()
    1

AUTHORS:

- William Stein (2005)

"""
# ****************************************************************************
#       Copyright (C) 2005 William Stein <wstein@gmail.com>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#  the License, or (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

from sage.misc.latex import latex

from sage.categories.finite_fields import FiniteFields
from sage.categories.fields import Fields

from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
from sage.schemes.generic.divisor_group import DivisorGroup
from sage.schemes.generic.divisor import Divisor_curve

from sage.rings.integer import Integer


class Curve_generic(AlgebraicScheme_subscheme):
    r"""
    Generic curve class.

    EXAMPLES::

        sage: A.<x,y,z> = AffineSpace(QQ, 3)
        sage: C = Curve([x - y, z - 2])
        sage: loads(C.dumps()) == C
        True
    """
    def _repr_(self):
        """
        Return a string representation of this curve.

        EXAMPLES::

            sage: A.<x,y,z> = AffineSpace(QQ, 3)
            sage: C = Curve([x - y, z - 2])
            sage: C
            Affine Curve over Rational Field defined by x - y, z - 2

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: C = Curve(x - y)
            sage: C
            Projective Plane Curve over Rational Field defined by x - y
        """
        if self.defining_ideal().is_zero() and self.ambient_space().dimension() == 1:
            return "{} Line over {}".format(self._repr_type(), self.base_ring())
        polys = ', '.join(str(x) for x in self.defining_polynomials())
        return "{} Curve over {} defined by {}".format(self._repr_type(),
                                                       self.base_ring(), polys)

    def _repr_type(self) -> str:
        r"""
        Return a string representation of the type of this curve.

        EXAMPLES::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = Curve([w^3 - z^3, w*x - x^2])
            sage: from sage.schemes.curves.curve import Curve_generic
            sage: Curve_generic._repr_type(C)
            'Generic'
        """
        return "Generic"

    def _latex_(self):
        r"""
        Return a latex representation of this curve.

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: latex(C)
            \text{Projective Plane curve over $\Bold{Q}$
            defined by $-x^{3} + y^{2} z - 17 x z^{2} + y z^{2}$}

            sage: A2 = AffineSpace(2, QQ, names=['x','y'])
            sage: x, y = A2.coordinate_ring().gens()
            sage: C = Curve(y^2 - x^3 - 17*x + y)
            sage: latex(C)
            \text{Affine Plane curve over $\Bold{Q}$
            defined by $-x^{3} + y^{2} - 17 x + y$}
        """
        if (self.defining_ideal().is_zero()
                and self.ambient_space().dimension() == 1):
            ambient_type, ring = self._repr_type(), latex(self.base_ring())
            return fr"\text{{{ambient_type} line over ${ring}$}}"
        else:
            ambient_type, ring = self._repr_type(), latex(self.base_ring())
            polys = ', '.join(f'${latex(p)}$' for p in self.defining_polynomials())
            return fr"\text{{{ambient_type} curve over ${ring}$ defined by {polys}}}"

    def dimension(self):
        r"""
        Return the dimension of the curve.

        Curves have dimension one by definition.

        EXAMPLES::

            sage: x = polygen(QQ)
            sage: C = HyperellipticCurve(x^7 + x^4 + x)
            sage: C.dimension()
            1
            sage: from sage.schemes.projective.projective_subscheme import AlgebraicScheme_subscheme_projective
            sage: AlgebraicScheme_subscheme_projective.dimension(C)
            1
        """
        return Integer(1)

    def defining_polynomial(self):
        """
        Return the defining polynomial of the curve.

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: C.defining_polynomial()
            -x^3 + y^2*z - 17*x*z^2 + y*z^2
        """
        return self.defining_polynomials()[0]

    def divisor_group(self, base_ring=None):
        r"""
        Return the divisor group of the curve.

        INPUT:

        - ``base_ring`` -- the base ring of the divisor group. Usually, this is
          `\ZZ` (default) or `\QQ`.

        OUTPUT: the divisor group of the curve

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C  = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: Cp = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: C.divisor_group() is Cp.divisor_group()
            True
        """
        return DivisorGroup(self, base_ring)

    def divisor(self, v, base_ring=None, check=True, reduce=True):
        r"""
        Return the divisor specified by ``v``.

        .. WARNING::

            The coefficients of the divisor must be in the base ring
            and the terms must be reduced. If you set ``check=False``
            and/or ``reduce=False`` it is your responsibility to pass
            a valid object ``v``.

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: p1 = C(0, -1, 1)
            sage: p2 = C(0, 0, 1)
            sage: p3 = C(0, 1, 0)
            sage: C.divisor([(1, p1), (-1, p2), (2, p3)])
            (x, y + z) - (x, y) + 2*(x, z)
        """
        return Divisor_curve(v, check=check, reduce=reduce, parent=self.divisor_group(base_ring))

    def genus(self):
        """
        Return the geometric genus of the curve.

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: C.genus()
            1
        """
        return self.geometric_genus()

    def geometric_genus(self):
        r"""
        Return the geometric genus of the curve.

        This is by definition the genus of the normalization of the projective
        closure of the curve over the algebraic closure of the base field; the
        base field must be a prime field.

        .. NOTE::

            This calls Singular's genus command.

        EXAMPLES:

        Examples of projective curves. ::

            sage: P2 = ProjectiveSpace(2, GF(5), names=['x','y','z'])
            sage: x, y, z = P2.coordinate_ring().gens()
            sage: C = Curve(y^2*z - x^3 - 17*x*z^2 + y*z^2)
            sage: C.geometric_genus()
            1
            sage: C = Curve(y^2*z - x^3)
            sage: C.geometric_genus()
            0
            sage: C = Curve(x^10 + y^7*z^3 + z^10)
            sage: C.geometric_genus()
            3

        Examples of affine curves. ::

            sage: x, y = PolynomialRing(GF(5), 2, 'xy').gens()
            sage: C = Curve(y^2 - x^3 - 17*x + y)
            sage: C.geometric_genus()
            1
            sage: C = Curve(y^2 - x^3)
            sage: C.geometric_genus()
            0
            sage: C = Curve(x^10 + y^7 + 1)
            sage: C.geometric_genus()
            3

        """
        try:
            return self._genus
        except AttributeError:
            self._genus = self.defining_ideal().genus()
            return self._genus

    def union(self, other):
        """
        Return the union of ``self`` and ``other``.

        EXAMPLES::

            sage: x,y,z = PolynomialRing(QQ, 3, names='x,y,z').gens()
            sage: C1 = Curve(z - x)
            sage: C2 = Curve(y - x)
            sage: C1.union(C2).defining_polynomial()
            x^2 - x*y - x*z + y*z
        """
        from .constructor import Curve
        return Curve(AlgebraicScheme_subscheme.union(self, other))

    __add__ = union

    def singular_subscheme(self):
        r"""
        Return the subscheme of singular points of this curve.

        OUTPUT:

        - a subscheme in the ambient space of this curve.

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(CC, 2)
            sage: C = Curve([y^4 - 2*x^5 - x^2*y], A)
            sage: C.singular_subscheme()
            Closed subscheme of Affine Space of dimension 2 over Complex Field
             with 53 bits of precision defined by:
              (-2.00000000000000)*x^5 + y^4 - x^2*y,
              (-10.0000000000000)*x^4 + (-2.00000000000000)*x*y,
              4.00000000000000*y^3 - x^2

        ::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = Curve([y^8 - x^2*z*w^5, w^2 - 2*y^2 - x*z], P)
            sage: C.singular_subscheme()
            Closed subscheme of Projective Space of dimension 3
             over Rational Field defined by:
              y^8 - x^2*z*w^5,
              -2*y^2 - x*z + w^2,
              -x^3*y*z^4 + 3*x^2*y*z^3*w^2 - 3*x*y*z^2*w^4 + 8*x*y*z*w^5 + y*z*w^6,
              x^2*z*w^5,
              -5*x^2*z^2*w^4 - 4*x*z*w^6,
              x^4*y*z^3 - 3*x^3*y*z^2*w^2 + 3*x^2*y*z*w^4 - 4*x^2*y*w^5 - x*y*w^6,
              -2*x^3*y*z^3*w + 6*x^2*y*z^2*w^3 - 20*x^2*y*z*w^4
               - 6*x*y*z*w^5 + 2*y*w^7,
              -5*x^3*z*w^4 - 2*x^2*w^6
        """
        return self.ambient_space().subscheme(self.Jacobian())

    def singular_points(self, F=None):
        r"""
        Return the set of singular points of this curve.

        INPUT:

        - ``F`` -- (default: None) field over which to find the singular
          points; if not given, the base ring of this curve is used

        OUTPUT: a list of points in the ambient space of this curve

        EXAMPLES::

            sage: A.<x,y,z> = AffineSpace(QQ, 3)
            sage: C = Curve([y^2 - x^5, x - z], A)
            sage: C.singular_points()
            [(0, 0, 0)]

        ::

            sage: # needs sage.rings.number_field
            sage: R.<a> = QQ[]
            sage: K.<b> = NumberField(a^8 - a^4 + 1)
            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: C = Curve([359/12*x*y^2*z^2 + 2*y*z^4 + 187/12*y^3*z^2 + x*z^4
            ....:            + 67/3*x^2*y*z^2 + 117/4*y^5 + 9*x^5 + 6*x^3*z^2
            ....:            + 393/4*x*y^4 + 145*x^2*y^3 + 115*x^3*y^2 + 49*x^4*y], P)
            sage: sorted(C.singular_points(K), key=str)
            [(-1/2*b^5 - 1/2*b^3 + 1/2*b - 1 : 1 : 0),
             (-2/3*b^4 + 1/3 : 0 : 1),
             (-b^6 : b^6 : 1),
             (1/2*b^5 + 1/2*b^3 - 1/2*b - 1 : 1 : 0),
             (2/3*b^4 - 1/3 : 0 : 1),
             (b^6 : -b^6 : 1)]
        """
        if F is None:
            if self.base_ring() not in Fields():
                raise TypeError("curve must be defined over a field")
            F = self.base_ring()
        elif F not in Fields():
            raise TypeError("(=%s) must be a field" % F)
        X = self.singular_subscheme()
        return [self.point(p, check=False) for p in X.rational_points(F=F)]

    def is_singular(self, P=None):
        r"""
        Return whether ``P`` is a singular point of this curve, or if no point
        is passed, whether this curve is singular or not.

        This just uses the is_smooth function for algebraic subschemes.

        INPUT:

        - ``P`` -- (default: None) a point on this curve

        OUTPUT:

        A boolean. If a point ``P`` is provided, and if ``P`` lies on this
        curve, returns True if ``P`` is a singular point of this curve, and
        False otherwise. If no point is provided, returns True or False
        depending on whether this curve is or is not singular, respectively.

        EXAMPLES::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = P.curve([y^2 - x^2 - z^2, z - w])
            sage: C.is_singular()
            False

        ::

            sage: A.<x,y,z> = AffineSpace(GF(11), 3)
            sage: C = A.curve([y^3 - z^5, x^5 - y + 1])
            sage: Q = A([7,0,0])
            sage: C.is_singular(Q)
            True
        """
        return not self.is_smooth(P)

    def intersects_at(self, C, P):
        r"""
        Return whether the point ``P`` is or is not in the intersection of this
        curve with the curve ``C``.

        INPUT:

        - ``C`` -- a curve in the same ambient space as this curve.

        - ``P`` -- a point in the ambient space of this curve.

        EXAMPLES::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = Curve([x^2 - z^2, y^3 - w*x^2], P)
            sage: D = Curve([w^2 - 2*x*y + z^2, y^2 - w^2], P)
            sage: Q1 = P([1,1,-1,1])
            sage: C.intersects_at(D, Q1)
            True
            sage: Q2 = P([0,0,1,-1])
            sage: C.intersects_at(D, Q2)
            False

        ::

            sage: A.<x,y> = AffineSpace(GF(13), 2)
            sage: C = Curve([y + 12*x^5 + 3*x^3 + 7], A)
            sage: D = Curve([y^2 + 7*x^2 + 8], A)
            sage: Q1 = A([9,6])
            sage: C.intersects_at(D, Q1)
            True
            sage: Q2 = A([3,7])
            sage: C.intersects_at(D, Q2)
            False
        """
        if C.ambient_space() != self.ambient_space():
            raise TypeError("(=%s) must be a curve in the same ambient space as (=%s)" % (C, self))
        if not isinstance(C, Curve_generic):
            raise TypeError("(=%s) must be a curve" % C)
        try:
            P = self.ambient_space()(P)
        except TypeError:
            raise TypeError("(=%s) must be a point in the ambient space of this curve" % P)
        try:
            P = self(P)
            P = C(P)
        except TypeError:
            return False
        return True

    def intersection_points(self, C, F=None):
        r"""
        Return the points in the intersection of this curve and the curve ``C``.

        If the intersection of these two curves has dimension greater than
        zero, and if the base ring of this curve is not a finite field, then an
        error is returned.

        INPUT:

        - ``C`` -- a curve in the same ambient space as this curve

        - ``F`` -- (default: None); field over which to compute the
          intersection points; if not specified, the base ring of this curve is
          used

        OUTPUT: a list of points in the ambient space of this curve

        EXAMPLES::

            sage: # needs sage.rings.number_field
            sage: R.<a> = QQ[]
            sage: K.<b> = NumberField(a^2 + a + 1)
            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = Curve([y^2 - w*z, w^3 - y^3], P)
            sage: D = Curve([x*y - w*z, z^3 - y^3], P)
            sage: C.intersection_points(D, F=K)
            [(-b - 1 : -b - 1 : b : 1), (b : b : -b - 1 : 1),
             (1 : 0 : 0 : 0), (1 : 1 : 1 : 1)]

        ::

            sage: A.<x,y> = AffineSpace(GF(7), 2)
            sage: C = Curve([y^3 - x^3], A)
            sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A)
            sage: C.intersection_points(D)
            [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 3), (5, 5), (5, 6), (6, 6)]

        ::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: C = Curve([y^3 - x^3], A)
            sage: D = Curve([-x*y^3 + y^4 - 2*x^3 + 2*x^2*y], A)
            sage: C.intersection_points(D)
            Traceback (most recent call last):
            ...
            NotImplementedError: the intersection must have dimension zero or
            (=Rational Field) must be a finite field
        """
        if C.ambient_space() != self.ambient_space():
            raise TypeError("(=%s) must be a curve in the same ambient space as (=%s)" % (C, self))
        if not isinstance(C, Curve_generic):
            raise TypeError("(=%s) must be a curve" % C)
        X = self.intersection(C)
        if F is None:
            F = self.base_ring()
        if X.dimension() == 0 or F in FiniteFields():
            return X.rational_points(F=F)
        else:
            raise NotImplementedError("the intersection must have dimension "
                                      "zero or (={}) must be a finite field".format(F))

    def change_ring(self, R):
        r"""
        Return a new curve which is this curve coerced to ``R``.

        INPUT:

        - ``R`` -- ring or embedding

        OUTPUT: a new curve which is this curve coerced to ``R``

        EXAMPLES::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
            sage: C = Curve([x^2 - y^2, z*y - 4/5*w^2], P)
            sage: C.change_ring(QuadraticField(-1))                                     # needs sage.rings.number_field
            Projective Curve over Number Field in a with defining polynomial x^2 + 1
             with a = 1*I defined by x^2 - y^2, y*z - 4/5*w^2

        ::

            sage: # needs sage.rings.number_field
            sage: R.<a> = QQ[]
            sage: K.<b> = NumberField(a^3 + a^2 - 1)
            sage: A.<x,y> = AffineSpace(K, 2)
            sage: C = Curve([K.0*x^2 - x + y^3 - 11], A)
            sage: L = K.embeddings(QQbar)
            sage: set_verbose(-1)  # suppress warnings for slow computation
            sage: C.change_ring(L[0])
            Affine Plane Curve over Algebraic Field defined
             by y^3 + (-0.8774388331233464? - 0.744861766619745?*I)*x^2 - x - 11

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: C = P.curve([y*x - 18*x^2 + 17*z^2])
            sage: C.change_ring(GF(17))
            Projective Plane Curve over Finite Field of size 17 defined by -x^2 + x*y
        """
        new_AS = self.ambient_space().change_ring(R)
        I = [f.change_ring(R) for f in self.defining_polynomials()]
        return new_AS.curve(I)
